Optimal pre-sale policy for deteriorating items

Lianxia Zhao; Hui Qiao; Qi An

doi:10.3934/naco.2021054

Article Contents

2022, Volume 12, Issue 1: 109-120. Doi: 10.3934/naco.2021054

This issue Previous Article Global optimality conditions and duality theorems for robust optimal solutions of optimization problems with data uncertainty, using underestimators Next Article Optimality and duality for complex multi-objective programming

Optimal pre-sale policy for deteriorating items

1.
School of Management, Shanghai University, Shanghai 200444, China
2.
School of Engineering, Open University of China, Beijing 100039, China

^* Corresponding author: Qi An
^* Corresponding author: Qi An

Received: May 2020

Revised: July 2021

Early access: November 2021

Published: March 2022

This work is supported by Humanities and Social Sciences Foundation of the Chinese Ministry of Education (20YJAZH135)and National Natural Science Foundation of China (71502100)

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Abstract

Pre-sale policy is a frequently-used sales approach for deteriorating products, e.g, fruits, vegetables, seafood, etc. In this paper, we consider an EOQ inventory model under pre-sale policy for deteriorating products, in which the demand of pre-sale period depends on price and pre-sale horizon, and the demand of spot-sale period depends on the price and stock level. Optimal pricing decisions and economic order quantity are also provided. We compare pre-sale model with a benchmark inventory model in which all the products are sold in spot-sale period. Theoretical results are derived to show the existence and uniqueness of the optimal solution. Numerical experiments are carried out to to illustrate the theoretical results. And sensitivity analysis is conducted to identify conditions under which the pre-sale policy is better off than the spot-sale only policy.

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Mathematics Subject Classification: Primary: 90B05, 90B60; Secondary: 91B42.

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Figure 1. The graphic of the sale model during a cycle

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Figure 2. Comparison of the profits and prices at different pre-sale market potential

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Figure 3. Comparison of the profits and prices at different deteriorating rate

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Figure 4. Comparison of the profits and prices at different price sensitivity

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Figure 5. Comparison of the profits and prices at different planning horizon

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Table 1. Summary of notations

Symbol	Description
$ \epsilon $	The market demand in pre-sale period
$ \delta $	The price sensitivity of the demand
$ T $	The fixed spot-sale planning horizon
$ p_0 $	The selling price of a unit in pre-sale period under pre-sale policy
$ p $	The selling price of a unit in spot-sale period under pre-sale policy
$ p_b $	The selling price of a unit under spot-sale only policy
$ w $	The purchase price of a unit
$ \theta $	The deterioration rate of the item
$ c $	The sensitivity to the advertising policy, $ 0\leq\delta\leq1 $
$ c_h $	The holding cost per unit per unit time
$ c_d $	The deterioration cost per unit per unit time
$ L $	The length of the pre-sale horizon, and suppose $ L=\epsilon T (0\leq\epsilon\leq\frac{1}{2}) $
$ I(t) $	The instantaneous inventory level on hand at time $ [0, T]. $

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